Well, I've now read in multiple places that the HTC One does not support rapid charging (
CLICK HERE).
I also downloaded an app called Battery Monitor Widget, which confirms as much. This app shows the net mA gain or loss at any given time. For example, when plugged into the AC adapter that came with the phone, it shows a positive 879mA. Since that adapter should charge at 1000mA, I assume the missing 121mA is due to the OS, the screen, and any running apps. I got the same results with a 2100mA wall charger (made by Rocketfish). When in the car, I noticed that the 3100mA charger (made by Rocketfish) never got any better results. My other car charger is 1000mA (made by Just Wireless, and is specifically designed for HTC phones), and again, got the same results. If I turned on navigation and music, the net gain was generally around just 100mA, so that's why charging takes so long in the car (again, recall that both my car chargers can charge in AC mode). I have not had the opportunity to drive for hours again with the navigation running, so the overheating issue is still up in the air, but I suspect the overheating caused the charging to stop, and that in turn caused my phone to have a net loss. In my shorter 30 minute drives today, the phone did not overheat and had a very slow net gain of about 1%.
From what I can tell, the bottom line is that the best you're going to ever get is 1000mA (i.e., 1 amp). HTC One owners get no benefit from a 2.1 amp or 3.1 amp charger. Because of that, and the fact that navigation takes so much battery power (particularly if it's downloading map data over a cellular network), the HTC One simply cannot charge at acceptable rates when navigating over long distances in the car.
Hopefully I can prevent overheating by directing a vent on the phone while driving. I don't know what I'll do in the winter, however, when I won't have my AC running.